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The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy ..."
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Cited by 149 (51 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Performance Modeling of Distributed Memory Architectures
 Journal of Parallel and Distributed Computing
, 1991
"... We provide performance models for several primitive operations on data structures distributed over memory units interconnected by a Boolean cube network. In particular, we model single source, and multiple source concurrent broadcasting or reduction, concurrent gather and scatter operations, shifts ..."
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Cited by 20 (7 self)
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We provide performance models for several primitive operations on data structures distributed over memory units interconnected by a Boolean cube network. In particular, we model single source, and multiple source concurrent broadcasting or reduction, concurrent gather and scatter operations, shifts along several axes of multidimensional arrays, and emulation of butterfly networks. We also show how the processor configuration, data aggregation, and the encoding of the address space affect the performance for two important basic computations: the multiplication of arbitrarily shaped matrices, and the Fast Fourier Transform. We also give an example of the performance behavior for local matrix operations for a processor with a single path to local memory, and a set of registers. The analytic models are verified by measurements on the Connection Machine model CM2. 1 Introduction This paper addresses crucial issues in performance modeling of distributed memory architectures designed for s...
Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique
, 1994
"... We study a new method in reducing the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Kosloff and TalEzer, and the proper choice of the parameter ff, the roundoff error of the kth derivative can be reduced from O(N 2k ) to O((N jln ..."
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Cited by 16 (0 self)
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We study a new method in reducing the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Kosloff and TalEzer, and the proper choice of the parameter ff, the roundoff error of the kth derivative can be reduced from O(N 2k ) to O((N jln fflj) k ), where ffl is the machine precision and N is the number of collocation points. This drastic reduction of roundoff error makes mapped Chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large N . We also study several other aspects of the mapped Chebyshev differentiation matrix. We find that 1) the mapped Chebyshev methods requires much less than ß points to resolve a wave, 2) the eigenvalues are less sensitive to perturbation by roundoff error, and 3) larger time steps can be used for solving PDEs. All these advantages of the mapped Chebyshev methods can be achieved while maintaining spectral accuracy. 1 Introduction In [5], we...
Parallel Fast Fourier Transforms for Non Power of Two Data
"... This report deals with parallel algorithms for computing discrete Fourier transforms of real sequences of length N not equal to a power of two. The method described is an extension of existing power of two transforms to sequences with N a product of small primes. In particular, this implementation r ..."
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This report deals with parallel algorithms for computing discrete Fourier transforms of real sequences of length N not equal to a power of two. The method described is an extension of existing power of two transforms to sequences with N a product of small primes. In particular, this implementation requires N = 2 p 3 q 5 r . The communication required is the same as for a transform of length N = 2 p . The algorithm presented is intended for use in the solution of partial differential equations, or in any situation in which a large number of forward and backward transforms must be performed and in which the Fourier Coefficients need not be ordered. This implementation is a one dimensional FFT but the techniques are applicable to multidimensional transforms as well. The algorithm has been implemented on a 128 node Intel Ipsc/860. 1 Introduction The algorithm described in this report is an extension of the parallel algorithm given by Swarztrauber [2] and Walker [3] for computing ...